LMFDB - Embedded newform 672.2.h.a.575.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(575,672)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("672.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	672.575
Dual form			672.2.h.a.575.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.292893 - 1.70711i) q^{3} +3.41421i q^{5} -1.00000i q^{7} +(-2.82843 + 1.00000i) q^{9} +O(q^{10})$	\(q+(-0.292893 - 1.70711i) q^{3} +3.41421i q^{5} -1.00000i q^{7} +(-2.82843 + 1.00000i) q^{9} -2.00000 q^{11} +3.41421 q^{13} +(5.82843 - 1.00000i) q^{15} +4.82843i q^{17} +6.24264i q^{19} +(-1.70711 + 0.292893i) q^{21} +4.00000 q^{23} -6.65685 q^{25} +(2.53553 + 4.53553i) q^{27} +4.82843i q^{29} -1.17157i q^{31} +(0.585786 + 3.41421i) q^{33} +3.41421 q^{35} -0.828427 q^{37} +(-1.00000 - 5.82843i) q^{39} -10.0000i q^{41} +10.4853i q^{43} +(-3.41421 - 9.65685i) q^{45} -1.65685 q^{47} -1.00000 q^{49} +(8.24264 - 1.41421i) q^{51} +13.3137i q^{53} -6.82843i q^{55} +(10.6569 - 1.82843i) q^{57} +6.24264 q^{59} +14.2426 q^{61} +(1.00000 + 2.82843i) q^{63} +11.6569i q^{65} -9.31371i q^{67} +(-1.17157 - 6.82843i) q^{69} +0.343146 q^{71} -3.17157 q^{73} +(1.94975 + 11.3640i) q^{75} +2.00000i q^{77} -4.00000i q^{79} +(7.00000 - 5.65685i) q^{81} -11.4142 q^{83} -16.4853 q^{85} +(8.24264 - 1.41421i) q^{87} +7.65685i q^{89} -3.41421i q^{91} +(-2.00000 + 0.343146i) q^{93} -21.3137 q^{95} +5.31371 q^{97} +(5.65685 - 2.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3}+O(q^{10}) $ 4 * q - 4 * q^3	$ 4 q - 4 q^{3} - 8 q^{11} + 8 q^{13} + 12 q^{15} - 4 q^{21} + 16 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{35} + 8 q^{37} - 4 q^{39} - 8 q^{45} + 16 q^{47} - 4 q^{49} + 16 q^{51} + 20 q^{57} + 8 q^{59} + 40 q^{61} + 4 q^{63} - 16 q^{69} + 24 q^{71} - 24 q^{73} - 12 q^{75} + 28 q^{81} - 40 q^{83} - 32 q^{85} + 16 q^{87} - 8 q^{93} - 40 q^{95} - 24 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^11 + 8 * q^13 + 12 * q^15 - 4 * q^21 + 16 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^33 + 8 * q^35 + 8 * q^37 - 4 * q^39 - 8 * q^45 + 16 * q^47 - 4 * q^49 + 16 * q^51 + 20 * q^57 + 8 * q^59 + 40 * q^61 + 4 * q^63 - 16 * q^69 + 24 * q^71 - 24 * q^73 - 12 * q^75 + 28 * q^81 - 40 * q^83 - 32 * q^85 + 16 * q^87 - 8 * q^93 - 40 * q^95 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.292893	−	1.70711i	−0.169102	−	0.985599i
$3$	−0.292893	−	1.70711i	−0.169102	−	0.985599i
$4$		0			0
$5$			3.41421i			1.52688i	0.645877	+	0.763441i	$0.276492\pi$
$5$			3.41421i			1.52688i	−0.645877	+	0.763441i	$0.723508\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−2.82843	+	1.00000i	−0.942809	+	0.333333i
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$	3.41421			0.946932			0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421			0.946932			0.473466	+	0.880812i	$0.343003\pi$
$14$		0			0
$15$	5.82843	−	1.00000i	1.50489	−	0.258199i
$16$		0			0
$17$			4.82843i			1.17107i	0.810649	+	0.585533i	$0.199115\pi$
$17$			4.82843i			1.17107i	−0.810649	+	0.585533i	$0.800885\pi$
$18$		0			0
$19$			6.24264i			1.43216i	0.698018	+	0.716080i	$0.254065\pi$
$19$			6.24264i			1.43216i	−0.698018	+	0.716080i	$0.745935\pi$
$20$		0			0
$21$	−1.70711	+	0.292893i	−0.372521	+	0.0639145i
$22$		0			0
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$		0			0
$25$	−6.65685			−1.33137
$26$		0			0
$27$	2.53553	+	4.53553i	0.487964	+	0.872864i
$28$		0			0
$29$			4.82843i			0.896616i	0.893879	+	0.448308i	$0.147973\pi$
$29$			4.82843i			0.896616i	−0.893879	+	0.448308i	$0.852027\pi$
$30$		0			0
$31$		−	1.17157i		−	0.210421i	−0.994450	−	0.105210i	$-0.966448\pi$
$31$		−	1.17157i		−	0.210421i	0.994450	−	0.105210i	$-0.0335516\pi$
$32$		0			0
$33$	0.585786	+	3.41421i	0.101972	+	0.594338i
$34$		0			0
$35$	3.41421			0.577107
$36$		0			0
$37$	−0.828427			−0.136193			−0.0680963	−	0.997679i	$-0.521693\pi$
$37$	−0.828427			−0.136193			−0.0680963	+	0.997679i	$0.521693\pi$
$38$		0			0
$39$	−1.00000	−	5.82843i	−0.160128	−	0.933295i
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$			10.4853i			1.59899i	0.600672	+	0.799495i	$0.294900\pi$
$43$			10.4853i			1.59899i	−0.600672	+	0.799495i	$0.705100\pi$
$44$		0			0
$45$	−3.41421	−	9.65685i	−0.508961	−	1.43956i
$46$		0			0
$47$	−1.65685			−0.241677			−0.120839	−	0.992672i	$-0.538558\pi$
$47$	−1.65685			−0.241677			−0.120839	+	0.992672i	$0.538558\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	8.24264	−	1.41421i	1.15420	−	0.198030i
$52$		0			0
$53$			13.3137i			1.82878i	0.404836	+	0.914389i	$0.367329\pi$
$53$			13.3137i			1.82878i	−0.404836	+	0.914389i	$0.632671\pi$
$54$		0			0
$55$		−	6.82843i		−	0.920745i
$56$		0			0
$57$	10.6569	−	1.82843i	1.41153	−	0.242181i
$58$		0			0
$59$	6.24264			0.812723			0.406361	−	0.913712i	$-0.366797\pi$
$59$	6.24264			0.812723			0.406361	+	0.913712i	$0.366797\pi$
$60$		0			0
$61$	14.2426			1.82358			0.911792	−	0.410653i	$-0.134699\pi$
$61$	14.2426			1.82358			0.911792	+	0.410653i	$0.134699\pi$
$62$		0			0
$63$	1.00000	+	2.82843i	0.125988	+	0.356348i
$64$		0			0
$65$			11.6569i			1.44585i
$66$		0			0
$67$		−	9.31371i		−	1.13785i	−0.822389	−	0.568925i	$-0.807359\pi$
$67$		−	9.31371i		−	1.13785i	0.822389	−	0.568925i	$-0.192641\pi$
$68$		0			0
$69$	−1.17157	−	6.82843i	−0.141041	−	0.822046i
$70$		0			0
$71$	0.343146			0.0407239			0.0203620	−	0.999793i	$-0.493518\pi$
$71$	0.343146			0.0407239			0.0203620	+	0.999793i	$0.493518\pi$
$72$		0			0
$73$	−3.17157			−0.371205			−0.185602	−	0.982625i	$-0.559424\pi$
$73$	−3.17157			−0.371205			−0.185602	+	0.982625i	$0.559424\pi$
$74$		0			0
$75$	1.94975	+	11.3640i	0.225137	+	1.31220i
$76$		0			0
$77$			2.00000i			0.227921i
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		0			0
$81$	7.00000	−	5.65685i	0.777778	−	0.628539i
$82$		0			0
$83$	−11.4142			−1.25287			−0.626436	−	0.779473i	$-0.715487\pi$
$83$	−11.4142			−1.25287			−0.626436	+	0.779473i	$0.715487\pi$
$84$		0			0
$85$	−16.4853			−1.78808
$86$		0			0
$87$	8.24264	−	1.41421i	0.883704	−	0.151620i
$88$		0			0
$89$			7.65685i			0.811625i	0.913956	+	0.405812i	$0.133011\pi$
$89$			7.65685i			0.811625i	−0.913956	+	0.405812i	$0.866989\pi$
$90$		0			0
$91$		−	3.41421i		−	0.357907i
$92$		0			0
$93$	−2.00000	+	0.343146i	−0.207390	+	0.0355826i
$94$		0			0
$95$	−21.3137			−2.18674
$96$		0			0
$97$	5.31371			0.539525			0.269763	−	0.962927i	$-0.413055\pi$
$97$	5.31371			0.539525			0.269763	+	0.962927i	$0.413055\pi$
$98$		0			0
$99$	5.65685	−	2.00000i	0.568535	−	0.201008i
$100$		0			0
$101$		−	18.2426i		−	1.81521i	−0.419825	−	0.907605i	$-0.637908\pi$
$101$		−	18.2426i		−	1.81521i	0.419825	−	0.907605i	$-0.362092\pi$
$102$		0			0
$103$		−	16.9706i		−	1.67216i	−0.548608	−	0.836080i	$-0.684842\pi$
$103$		−	16.9706i		−	1.67216i	0.548608	−	0.836080i	$-0.315158\pi$
$104$		0			0
$105$	−1.00000	−	5.82843i	−0.0975900	−	0.568796i
$106$		0			0
$107$	−14.4853			−1.40035			−0.700173	−	0.713974i	$-0.746894\pi$
$107$	−14.4853			−1.40035			−0.700173	+	0.713974i	$0.746894\pi$
$108$		0			0
$109$	−2.48528			−0.238047			−0.119023	−	0.992891i	$-0.537976\pi$
$109$	−2.48528			−0.238047			−0.119023	+	0.992891i	$0.537976\pi$
$110$		0			0
$111$	0.242641	+	1.41421i	0.0230304	+	0.134231i
$112$		0			0
$113$		−	5.31371i		−	0.499872i	−0.968262	−	0.249936i	$-0.919590\pi$
$113$		−	5.31371i		−	0.499872i	0.968262	−	0.249936i	$-0.0804095\pi$
$114$		0			0
$115$			13.6569i			1.27351i
$116$		0			0
$117$	−9.65685	+	3.41421i	−0.892776	+	0.315644i
$118$		0			0
$119$	4.82843			0.442621
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	−17.0711	+	2.92893i	−1.53925	+	0.264093i
$124$		0			0
$125$		−	5.65685i		−	0.505964i
$126$		0			0
$127$			2.00000i			0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$			2.00000i			0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$		0			0
$129$	17.8995	−	3.07107i	1.57596	−	0.270392i
$130$		0			0
$131$	−21.5563			−1.88339			−0.941693	−	0.336472i	$-0.890766\pi$
$131$	−21.5563			−1.88339			−0.941693	+	0.336472i	$0.890766\pi$
$132$		0			0
$133$	6.24264			0.541306
$134$		0			0
$135$	−15.4853	+	8.65685i	−1.33276	+	0.745063i
$136$		0			0
$137$			11.3137i			0.966595i	0.875456	+	0.483298i	$0.160561\pi$
$137$			11.3137i			0.966595i	−0.875456	+	0.483298i	$0.839439\pi$
$138$		0			0
$139$		−	1.75736i		−	0.149057i	−0.997219	−	0.0745286i	$-0.976255\pi$
$139$		−	1.75736i		−	0.149057i	0.997219	−	0.0745286i	$-0.0237452\pi$
$140$		0			0
$141$	0.485281	+	2.82843i	0.0408681	+	0.238197i
$142$		0			0
$143$	−6.82843			−0.571022
$144$		0			0
$145$	−16.4853			−1.36903
$146$		0			0
$147$	0.292893	+	1.70711i	0.0241574	+	0.140800i
$148$		0			0
$149$			7.65685i			0.627274i	0.949543	+	0.313637i	$0.101547\pi$
$149$			7.65685i			0.627274i	−0.949543	+	0.313637i	$0.898453\pi$
$150$		0			0
$151$			18.0000i			1.46482i	0.680864	+	0.732410i	$0.261604\pi$
$151$			18.0000i			1.46482i	−0.680864	+	0.732410i	$0.738396\pi$
$152$		0			0
$153$	−4.82843	−	13.6569i	−0.390355	−	1.10409i
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	2.24264			0.178982			0.0894911	−	0.995988i	$-0.471476\pi$
$157$	2.24264			0.178982			0.0894911	+	0.995988i	$0.471476\pi$
$158$		0			0
$159$	22.7279	−	3.89949i	1.80244	−	0.309250i
$160$		0			0
$161$		−	4.00000i		−	0.315244i
$162$		0			0
$163$		−	8.14214i		−	0.637741i	−0.947798	−	0.318871i	$-0.896696\pi$
$163$		−	8.14214i		−	0.637741i	0.947798	−	0.318871i	$-0.103304\pi$
$164$		0			0
$165$	−11.6569	+	2.00000i	−0.907485	+	0.155700i
$166$		0			0
$167$	10.8284			0.837929			0.418964	−	0.908003i	$-0.362393\pi$
$167$	10.8284			0.837929			0.418964	+	0.908003i	$0.362393\pi$
$168$		0			0
$169$	−1.34315			−0.103319
$170$		0			0
$171$	−6.24264	−	17.6569i	−0.477387	−	1.35025i
$172$		0			0
$173$		−	9.55635i		−	0.726556i	−0.931681	−	0.363278i	$-0.881658\pi$
$173$		−	9.55635i		−	0.726556i	0.931681	−	0.363278i	$-0.118342\pi$
$174$		0			0
$175$			6.65685i			0.503211i
$176$		0			0
$177$	−1.82843	−	10.6569i	−0.137433	−	0.801018i
$178$		0			0
$179$	−6.48528			−0.484733			−0.242366	−	0.970185i	$-0.577924\pi$
$179$	−6.48528			−0.484733			−0.242366	+	0.970185i	$0.577924\pi$
$180$		0			0
$181$	20.3848			1.51519			0.757594	−	0.652726i	$-0.226375\pi$
$181$	20.3848			1.51519			0.757594	+	0.652726i	$0.226375\pi$
$182$		0			0
$183$	−4.17157	−	24.3137i	−0.308372	−	1.79732i
$184$		0			0
$185$		−	2.82843i		−	0.207950i
$186$		0			0
$187$		−	9.65685i		−	0.706179i
$188$		0			0
$189$	4.53553	−	2.53553i	0.329912	−	0.184433i
$190$		0			0
$191$	3.65685			0.264601			0.132300	−	0.991210i	$-0.457764\pi$
$191$	3.65685			0.264601			0.132300	+	0.991210i	$0.457764\pi$
$192$		0			0
$193$	17.6569			1.27097			0.635484	−	0.772114i	$-0.280801\pi$
$193$	17.6569			1.27097			0.635484	+	0.772114i	$0.280801\pi$
$194$		0			0
$195$	19.8995	−	3.41421i	1.42503	−	0.244497i
$196$		0			0
$197$		−	26.0000i		−	1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$		−	26.0000i		−	1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$		0			0
$199$			7.31371i			0.518455i	0.965816	+	0.259228i	$0.0834680\pi$
$199$			7.31371i			0.518455i	−0.965816	+	0.259228i	$0.916532\pi$
$200$		0			0
$201$	−15.8995	+	2.72792i	−1.12146	+	0.192413i
$202$		0			0
$203$	4.82843			0.338889
$204$		0			0
$205$	34.1421			2.38459
$206$		0			0
$207$	−11.3137	+	4.00000i	−0.786357	+	0.278019i
$208$		0			0
$209$		−	12.4853i		−	0.863625i
$210$		0			0
$211$		−	0.343146i		−	0.0236231i	−0.999930	−	0.0118116i	$-0.996240\pi$
$211$		−	0.343146i		−	0.0236231i	0.999930	−	0.0118116i	$-0.00375982\pi$
$212$		0			0
$213$	−0.100505	−	0.585786i	−0.00688649	−	0.0401374i
$214$		0			0
$215$	−35.7990			−2.44147
$216$		0			0
$217$	−1.17157			−0.0795315
$218$		0			0
$219$	0.928932	+	5.41421i	0.0627714	+	0.365859i
$220$		0			0
$221$			16.4853i			1.10892i
$222$		0			0
$223$			2.82843i			0.189405i	0.995506	+	0.0947027i	$0.0301901\pi$
$223$			2.82843i			0.189405i	−0.995506	+	0.0947027i	$0.969810\pi$
$224$		0			0
$225$	18.8284	−	6.65685i	1.25523	−	0.443790i
$226$		0			0
$227$	23.8995			1.58627			0.793133	−	0.609049i	$-0.208449\pi$
$227$	23.8995			1.58627			0.793133	+	0.609049i	$0.208449\pi$
$228$		0			0
$229$	−1.07107			−0.0707782			−0.0353891	−	0.999374i	$-0.511267\pi$
$229$	−1.07107			−0.0707782			−0.0353891	+	0.999374i	$0.511267\pi$
$230$		0			0
$231$	3.41421	−	0.585786i	0.224639	−	0.0385419i
$232$		0			0
$233$			8.97056i			0.587681i	0.955854	+	0.293841i	$0.0949335\pi$
$233$			8.97056i			0.587681i	−0.955854	+	0.293841i	$0.905067\pi$
$234$		0			0
$235$		−	5.65685i		−	0.369012i
$236$		0			0
$237$	−6.82843	+	1.17157i	−0.443554	+	0.0761018i
$238$		0			0
$239$	11.3137			0.731823			0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137			0.731823			0.365911	+	0.930650i	$0.380757\pi$
$240$		0			0
$241$	3.65685			0.235559			0.117779	−	0.993040i	$-0.462422\pi$
$241$	3.65685			0.235559			0.117779	+	0.993040i	$0.462422\pi$
$242$		0			0
$243$	−11.7071	−	10.2929i	−0.751011	−	0.660289i
$244$		0			0
$245$		−	3.41421i		−	0.218126i
$246$		0			0
$247$			21.3137i			1.35616i
$248$		0			0
$249$	3.34315	+	19.4853i	0.211863	+	1.23483i
$250$		0			0
$251$	20.5858			1.29936			0.649682	−	0.760206i	$-0.274902\pi$
$251$	20.5858			1.29936			0.649682	+	0.760206i	$0.274902\pi$
$252$		0			0
$253$	−8.00000			−0.502956
$254$		0			0
$255$	4.82843	+	28.1421i	0.302368	+	1.76233i
$256$		0			0
$257$		−	14.4853i		−	0.903567i	−0.892128	−	0.451784i	$-0.850788\pi$
$257$		−	14.4853i		−	0.903567i	0.892128	−	0.451784i	$-0.149212\pi$
$258$		0			0
$259$			0.828427i			0.0514760i
$260$		0			0
$261$	−4.82843	−	13.6569i	−0.298872	−	0.845338i
$262$		0			0
$263$	1.31371			0.0810067			0.0405034	−	0.999179i	$-0.487104\pi$
$263$	1.31371			0.0810067			0.0405034	+	0.999179i	$0.487104\pi$
$264$		0			0
$265$	−45.4558			−2.79233
$266$		0			0
$267$	13.0711	−	2.24264i	0.799936	−	0.137247i
$268$		0			0
$269$			18.7279i			1.14186i	0.820998	+	0.570931i	$0.193418\pi$
$269$			18.7279i			1.14186i	−0.820998	+	0.570931i	$0.806582\pi$
$270$		0			0
$271$		−	12.4853i		−	0.758427i	−0.925309	−	0.379213i	$-0.876195\pi$
$271$		−	12.4853i		−	0.758427i	0.925309	−	0.379213i	$-0.123805\pi$
$272$		0			0
$273$	−5.82843	+	1.00000i	−0.352752	+	0.0605228i
$274$		0			0
$275$	13.3137			0.802847
$276$		0			0
$277$	−1.31371			−0.0789331			−0.0394665	−	0.999221i	$-0.512566\pi$
$277$	−1.31371			−0.0789331			−0.0394665	+	0.999221i	$0.512566\pi$
$278$		0			0
$279$	1.17157	+	3.31371i	0.0701402	+	0.198387i
$280$		0			0
$281$		−	8.97056i		−	0.535139i	−0.963539	−	0.267569i	$-0.913780\pi$
$281$		−	8.97056i		−	0.535139i	0.963539	−	0.267569i	$-0.0862205\pi$
$282$		0			0
$283$		−	10.2426i		−	0.608862i	−0.952534	−	0.304431i	$-0.901534\pi$
$283$		−	10.2426i		−	0.608862i	0.952534	−	0.304431i	$-0.0984663\pi$
$284$		0			0
$285$	6.24264	+	36.3848i	0.369782	+	2.15525i
$286$		0			0
$287$	−10.0000			−0.590281
$288$		0			0
$289$	−6.31371			−0.371395
$290$		0			0
$291$	−1.55635	−	9.07107i	−0.0912348	−	0.531755i
$292$		0			0
$293$			1.27208i			0.0743156i	0.999309	+	0.0371578i	$0.0118304\pi$
$293$			1.27208i			0.0743156i	−0.999309	+	0.0371578i	$0.988170\pi$
$294$		0			0
$295$			21.3137i			1.24093i
$296$		0			0
$297$	−5.07107	−	9.07107i	−0.294253	−	0.526357i
$298$		0			0
$299$	13.6569			0.789796
$300$		0			0
$301$	10.4853			0.604362
$302$		0			0
$303$	−31.1421	+	5.34315i	−1.78907	+	0.306956i
$304$		0			0
$305$			48.6274i			2.78440i
$306$		0			0
$307$			9.55635i			0.545410i	0.962098	+	0.272705i	$0.0879182\pi$
$307$			9.55635i			0.545410i	−0.962098	+	0.272705i	$0.912082\pi$
$308$		0			0
$309$	−28.9706	+	4.97056i	−1.64808	+	0.282765i
$310$		0			0
$311$	19.7990			1.12270			0.561349	−	0.827579i	$-0.310283\pi$
$311$	19.7990			1.12270			0.561349	+	0.827579i	$0.310283\pi$
$312$		0			0
$313$	−8.82843			−0.499012			−0.249506	−	0.968373i	$-0.580268\pi$
$313$	−8.82843			−0.499012			−0.249506	+	0.968373i	$0.580268\pi$
$314$		0			0
$315$	−9.65685	+	3.41421i	−0.544102	+	0.192369i
$316$		0			0
$317$			10.0000i			0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$			10.0000i			0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$		0			0
$319$		−	9.65685i		−	0.540680i
$320$		0			0
$321$	4.24264	+	24.7279i	0.236801	+	1.38018i
$322$		0			0
$323$	−30.1421			−1.67715
$324$		0			0
$325$	−22.7279			−1.26072
$326$		0			0
$327$	0.727922	+	4.24264i	0.0402542	+	0.234619i
$328$		0			0
$329$			1.65685i			0.0913453i
$330$		0			0
$331$		−	4.82843i		−	0.265394i	−0.991157	−	0.132697i	$-0.957636\pi$
$331$		−	4.82843i		−	0.265394i	0.991157	−	0.132697i	$-0.0423638\pi$
$332$		0			0
$333$	2.34315	−	0.828427i	0.128404	−	0.0453975i
$334$		0			0
$335$	31.7990			1.73736
$336$		0			0
$337$	27.3137			1.48787			0.743936	−	0.668251i	$-0.232957\pi$
$337$	27.3137			1.48787			0.743936	+	0.668251i	$0.232957\pi$
$338$		0			0
$339$	−9.07107	+	1.55635i	−0.492673	+	0.0845293i
$340$		0			0
$341$			2.34315i			0.126888i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	23.3137	−	4.00000i	1.25517	−	0.215353i
$346$		0			0
$347$	−0.343146			−0.0184210			−0.00921051	−	0.999958i	$-0.502932\pi$
$347$	−0.343146			−0.0184210			−0.00921051	+	0.999958i	$0.502932\pi$
$348$		0			0
$349$	14.9289			0.799127			0.399564	−	0.916705i	$-0.369162\pi$
$349$	14.9289			0.799127			0.399564	+	0.916705i	$0.369162\pi$
$350$		0			0
$351$	8.65685	+	15.4853i	0.462069	+	0.826543i
$352$		0			0
$353$		−	19.4558i		−	1.03553i	−0.855523	−	0.517765i	$-0.826764\pi$
$353$		−	19.4558i		−	1.03553i	0.855523	−	0.517765i	$-0.173236\pi$
$354$		0			0
$355$			1.17157i			0.0621806i
$356$		0			0
$357$	−1.41421	−	8.24264i	−0.0748481	−	0.436247i
$358$		0			0
$359$	26.6274			1.40534			0.702671	−	0.711515i	$-0.251991\pi$
$359$	26.6274			1.40534			0.702671	+	0.711515i	$0.251991\pi$
$360$		0			0
$361$	−19.9706			−1.05108
$362$		0			0
$363$	2.05025	+	11.9497i	0.107610	+	0.627199i
$364$		0			0
$365$		−	10.8284i		−	0.566786i
$366$		0			0
$367$		−	13.1716i		−	0.687551i	−0.939052	−	0.343775i	$-0.888294\pi$
$367$		−	13.1716i		−	0.687551i	0.939052	−	0.343775i	$-0.111706\pi$
$368$		0			0
$369$	10.0000	+	28.2843i	0.520579	+	1.47242i
$370$		0			0
$371$	13.3137			0.691213
$372$		0			0
$373$	−22.9706			−1.18937			−0.594685	−	0.803959i	$-0.702723\pi$
$373$	−22.9706			−1.18937			−0.594685	+	0.803959i	$0.702723\pi$
$374$		0			0
$375$	−9.65685	+	1.65685i	−0.498678	+	0.0855596i
$376$		0			0
$377$			16.4853i			0.849035i
$378$		0			0
$379$		−	34.4853i		−	1.77139i	−0.464268	−	0.885695i	$-0.653682\pi$
$379$		−	34.4853i		−	1.77139i	0.464268	−	0.885695i	$-0.346318\pi$
$380$		0			0
$381$	3.41421	−	0.585786i	0.174915	−	0.0300107i
$382$		0			0
$383$	−5.65685			−0.289052			−0.144526	−	0.989501i	$-0.546166\pi$
$383$	−5.65685			−0.289052			−0.144526	+	0.989501i	$0.546166\pi$
$384$		0			0
$385$	−6.82843			−0.348009
$386$		0			0
$387$	−10.4853	−	29.6569i	−0.532997	−	1.50754i
$388$		0			0
$389$			6.48528i			0.328817i	0.986392	+	0.164408i	$0.0525715\pi$
$389$			6.48528i			0.328817i	−0.986392	+	0.164408i	$0.947429\pi$
$390$		0			0
$391$			19.3137i			0.976736i
$392$		0			0
$393$	6.31371	+	36.7990i	0.318484	+	1.85626i
$394$		0			0
$395$	13.6569			0.687151
$396$		0			0
$397$	18.2426			0.915572			0.457786	−	0.889062i	$-0.348643\pi$
$397$	18.2426			0.915572			0.457786	+	0.889062i	$0.348643\pi$
$398$		0			0
$399$	−1.82843	−	10.6569i	−0.0915358	−	0.533510i
$400$		0			0
$401$		−	31.6569i		−	1.58087i	−0.612547	−	0.790434i	$-0.709855\pi$
$401$		−	31.6569i		−	1.58087i	0.612547	−	0.790434i	$-0.290145\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$		0			0
$405$	19.3137	+	23.8995i	0.959706	+	1.18758i
$406$		0			0
$407$	1.65685			0.0821272
$408$		0			0
$409$	26.4853			1.30961			0.654806	−	0.755797i	$-0.272750\pi$
$409$	26.4853			1.30961			0.654806	+	0.755797i	$0.272750\pi$
$410$		0			0
$411$	19.3137	−	3.31371i	0.952675	−	0.163453i
$412$		0			0
$413$		−	6.24264i		−	0.307180i
$414$		0			0
$415$		−	38.9706i		−	1.91299i
$416$		0			0
$417$	−3.00000	+	0.514719i	−0.146911	+	0.0252059i
$418$		0			0
$419$	18.7279			0.914919			0.457459	−	0.889230i	$-0.348759\pi$
$419$	18.7279			0.914919			0.457459	+	0.889230i	$0.348759\pi$
$420$		0			0
$421$	−15.6569			−0.763068			−0.381534	−	0.924355i	$-0.624604\pi$
$421$	−15.6569			−0.763068			−0.381534	+	0.924355i	$0.624604\pi$
$422$		0			0
$423$	4.68629	−	1.65685i	0.227855	−	0.0805590i
$424$		0			0
$425$		−	32.1421i		−	1.55912i
$426$		0			0
$427$		−	14.2426i		−	0.689250i
$428$		0			0
$429$	2.00000	+	11.6569i	0.0965609	+	0.562798i
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$	14.9706			0.719439			0.359720	−	0.933060i	$-0.382872\pi$
$433$	14.9706			0.719439			0.359720	+	0.933060i	$0.382872\pi$
$434$		0			0
$435$	4.82843	+	28.1421i	0.231505	+	1.34931i
$436$		0			0
$437$			24.9706i			1.19450i
$438$		0			0
$439$			40.2843i			1.92266i	0.275393	+	0.961332i	$0.411192\pi$
$439$			40.2843i			1.92266i	−0.275393	+	0.961332i	$0.588808\pi$
$440$		0			0
$441$	2.82843	−	1.00000i	0.134687	−	0.0476190i
$442$		0			0
$443$	−23.1716			−1.10091			−0.550457	−	0.834863i	$-0.685547\pi$
$443$	−23.1716			−1.10091			−0.550457	+	0.834863i	$0.685547\pi$
$444$		0			0
$445$	−26.1421			−1.23926
$446$		0			0
$447$	13.0711	−	2.24264i	0.618240	−	0.106073i
$448$		0			0
$449$			18.3431i			0.865667i	0.901474	+	0.432833i	$0.142486\pi$
$449$			18.3431i			0.865667i	−0.901474	+	0.432833i	$0.857514\pi$
$450$		0			0
$451$			20.0000i			0.941763i
$452$		0			0
$453$	30.7279	−	5.27208i	1.44372	−	0.247704i
$454$		0			0
$455$	11.6569			0.546482
$456$		0			0
$457$	−24.0000			−1.12267			−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000			−1.12267			−0.561336	+	0.827588i	$0.689713\pi$
$458$		0			0
$459$	−21.8995	+	12.2426i	−1.02218	+	0.571438i
$460$		0			0
$461$		−	22.0416i		−	1.02658i	−0.858215	−	0.513291i	$-0.828426\pi$
$461$		−	22.0416i		−	1.02658i	0.858215	−	0.513291i	$-0.171574\pi$
$462$		0			0
$463$		−	1.65685i		−	0.0770005i	−0.999259	−	0.0385003i	$-0.987742\pi$
$463$		−	1.65685i		−	0.0770005i	0.999259	−	0.0385003i	$-0.0122580\pi$
$464$		0			0
$465$	−1.17157	−	6.82843i	−0.0543304	−	0.316661i
$466$		0			0
$467$	−18.0416			−0.834867			−0.417434	−	0.908707i	$-0.637070\pi$
$467$	−18.0416			−0.834867			−0.417434	+	0.908707i	$0.637070\pi$
$468$		0			0
$469$	−9.31371			−0.430067
$470$		0			0
$471$	−0.656854	−	3.82843i	−0.0302662	−	0.176405i
$472$		0			0
$473$		−	20.9706i		−	0.964228i
$474$		0			0
$475$		−	41.5563i		−	1.90674i
$476$		0			0
$477$	−13.3137	−	37.6569i	−0.609593	−	1.72419i
$478$		0			0
$479$	11.3137			0.516937			0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137			0.516937			0.258468	+	0.966020i	$0.416782\pi$
$480$		0			0
$481$	−2.82843			−0.128965
$482$		0			0
$483$	−6.82843	+	1.17157i	−0.310704	+	0.0533084i
$484$		0			0
$485$			18.1421i			0.823792i
$486$		0			0
$487$			22.0000i			0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$			22.0000i			0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$		0			0
$489$	−13.8995	+	2.38478i	−0.628557	+	0.107843i
$490$		0			0
$491$	7.17157			0.323649			0.161824	−	0.986820i	$-0.448262\pi$
$491$	7.17157			0.323649			0.161824	+	0.986820i	$0.448262\pi$
$492$		0			0
$493$	−23.3137			−1.05000
$494$		0			0
$495$	6.82843	+	19.3137i	0.306915	+	0.868087i
$496$		0			0
$497$		−	0.343146i		−	0.0153922i
$498$		0			0
$499$			36.6274i			1.63967i	0.572601	+	0.819834i	$0.305934\pi$
$499$			36.6274i			1.63967i	−0.572601	+	0.819834i	$0.694066\pi$
$500$		0			0
$501$	−3.17157	−	18.4853i	−0.141695	−	0.825861i
$502$		0			0
$503$	−25.4558			−1.13502			−0.567510	−	0.823367i	$-0.692093\pi$
$503$	−25.4558			−1.13502			−0.567510	+	0.823367i	$0.692093\pi$
$504$		0			0
$505$	62.2843			2.77161
$506$		0			0
$507$	0.393398	+	2.29289i	0.0174714	+	0.101831i
$508$		0			0
$509$		−	20.8701i		−	0.925049i	−0.886606	−	0.462525i	$-0.846944\pi$
$509$		−	20.8701i		−	0.925049i	0.886606	−	0.462525i	$-0.153056\pi$
$510$		0			0
$511$			3.17157i			0.140302i
$512$		0			0
$513$	−28.3137	+	15.8284i	−1.25008	+	0.698842i
$514$		0			0
$515$	57.9411			2.55319
$516$		0			0
$517$	3.31371			0.145737
$518$		0			0
$519$	−16.3137	+	2.79899i	−0.716092	+	0.122862i
$520$		0			0
$521$		−	12.3431i		−	0.540763i	−0.962753	−	0.270382i	$-0.912850\pi$
$521$		−	12.3431i		−	0.540763i	0.962753	−	0.270382i	$-0.0871498\pi$
$522$		0			0
$523$			39.8995i			1.74468i	0.488897	+	0.872342i	$0.337399\pi$
$523$			39.8995i			1.74468i	−0.488897	+	0.872342i	$0.662601\pi$
$524$		0			0
$525$	11.3640	−	1.94975i	0.495964	−	0.0850940i
$526$		0			0
$527$	5.65685			0.246416
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$	−17.6569	+	6.24264i	−0.766242	+	0.270908i
$532$		0			0
$533$		−	34.1421i		−	1.47886i
$534$		0			0
$535$		−	49.4558i		−	2.13816i
$536$		0			0
$537$	1.89949	+	11.0711i	0.0819693	+	0.477752i
$538$		0			0
$539$	2.00000			0.0861461
$540$		0			0
$541$	12.6274			0.542895			0.271448	−	0.962453i	$-0.412498\pi$
$541$	12.6274			0.542895			0.271448	+	0.962453i	$0.412498\pi$
$542$		0			0
$543$	−5.97056	−	34.7990i	−0.256221	−	1.49337i
$544$		0			0
$545$		−	8.48528i		−	0.363470i
$546$		0			0
$547$		−	8.14214i		−	0.348133i	−0.984734	−	0.174066i	$-0.944309\pi$
$547$		−	8.14214i		−	0.348133i	0.984734	−	0.174066i	$-0.0556907\pi$
$548$		0			0
$549$	−40.2843	+	14.2426i	−1.71929	+	0.607861i
$550$		0			0
$551$	−30.1421			−1.28410
$552$		0			0
$553$	−4.00000			−0.170097
$554$		0			0
$555$	−4.82843	+	0.828427i	−0.204955	+	0.0351648i
$556$		0			0
$557$		−	6.68629i		−	0.283307i	−0.989916	−	0.141654i	$-0.954758\pi$
$557$		−	6.68629i		−	0.283307i	0.989916	−	0.141654i	$-0.0452419\pi$
$558$		0			0
$559$			35.7990i			1.51414i
$560$		0			0
$561$	−16.4853	+	2.82843i	−0.696009	+	0.119416i
$562$		0			0
$563$	7.21320			0.304000			0.152000	−	0.988380i	$-0.451429\pi$
$563$	7.21320			0.304000			0.152000	+	0.988380i	$0.451429\pi$
$564$		0			0
$565$	18.1421			0.763245
$566$		0			0
$567$	−5.65685	−	7.00000i	−0.237566	−	0.293972i
$568$		0			0
$569$		−	1.31371i		−	0.0550735i	−0.999621	−	0.0275368i	$-0.991234\pi$
$569$		−	1.31371i		−	0.0550735i	0.999621	−	0.0275368i	$-0.00876633\pi$
$570$		0			0
$571$			8.14214i			0.340738i	0.985380	+	0.170369i	$0.0544959\pi$
$571$			8.14214i			0.340738i	−0.985380	+	0.170369i	$0.945504\pi$
$572$		0			0
$573$	−1.07107	−	6.24264i	−0.0447445	−	0.260790i
$574$		0			0
$575$	−26.6274			−1.11044
$576$		0			0
$577$	29.3137			1.22035			0.610173	−	0.792268i	$-0.291100\pi$
$577$	29.3137			1.22035			0.610173	+	0.792268i	$0.291100\pi$
$578$		0			0
$579$	−5.17157	−	30.1421i	−0.214923	−	1.25266i
$580$		0			0
$581$			11.4142i			0.473541i
$582$		0			0
$583$		−	26.6274i		−	1.10279i
$584$		0			0
$585$	−11.6569	−	32.9706i	−0.481952	−	1.36316i
$586$		0			0
$587$	−30.7279			−1.26828			−0.634139	−	0.773219i	$-0.718645\pi$
$587$	−30.7279			−1.26828			−0.634139	+	0.773219i	$0.718645\pi$
$588$		0			0
$589$	7.31371			0.301356
$590$		0			0
$591$	−44.3848	+	7.61522i	−1.82575	+	0.313248i
$592$		0			0
$593$		−	22.4853i		−	0.923360i	−0.887047	−	0.461680i	$-0.847247\pi$
$593$		−	22.4853i		−	0.923360i	0.887047	−	0.461680i	$-0.152753\pi$
$594$		0			0
$595$			16.4853i			0.675831i
$596$		0			0
$597$	12.4853	−	2.14214i	0.510989	−	0.0876718i
$598$		0			0
$599$	16.6274			0.679378			0.339689	−	0.940538i	$-0.389678\pi$
$599$	16.6274			0.679378			0.339689	+	0.940538i	$0.389678\pi$
$600$		0			0
$601$	−16.1421			−0.658451			−0.329226	−	0.944251i	$-0.606788\pi$
$601$	−16.1421			−0.658451			−0.329226	+	0.944251i	$0.606788\pi$
$602$		0			0
$603$	9.31371	+	26.3431i	0.379284	+	1.07278i
$604$		0			0
$605$		−	23.8995i		−	0.971653i
$606$		0			0
$607$			2.14214i			0.0869466i	0.999055	+	0.0434733i	$0.0138423\pi$
$607$			2.14214i			0.0869466i	−0.999055	+	0.0434733i	$0.986158\pi$
$608$		0			0
$609$	−1.41421	−	8.24264i	−0.0573068	−	0.334009i
$610$		0			0
$611$	−5.65685			−0.228852
$612$		0			0
$613$	26.7696			1.08121			0.540606	−	0.841276i	$-0.318195\pi$
$613$	26.7696			1.08121			0.540606	+	0.841276i	$0.318195\pi$
$614$		0			0
$615$	−10.0000	−	58.2843i	−0.403239	−	2.35025i
$616$		0			0
$617$		−	28.6274i		−	1.15250i	−0.817275	−	0.576248i	$-0.804516\pi$
$617$		−	28.6274i		−	1.15250i	0.817275	−	0.576248i	$-0.195484\pi$
$618$		0			0
$619$			18.2426i			0.733234i	0.930372	+	0.366617i	$0.119484\pi$
$619$			18.2426i			0.733234i	−0.930372	+	0.366617i	$0.880516\pi$
$620$		0			0
$621$	10.1421	+	18.1421i	0.406990	+	0.728019i
$622$		0			0
$623$	7.65685			0.306765
$624$		0			0
$625$	−13.9706			−0.558823
$626$		0			0
$627$	−21.3137	+	3.65685i	−0.851188	+	0.146041i
$628$		0			0
$629$		−	4.00000i		−	0.159490i
$630$		0			0
$631$		−	8.97056i		−	0.357112i	−0.983930	−	0.178556i	$-0.942857\pi$
$631$		−	8.97056i		−	0.357112i	0.983930	−	0.178556i	$-0.0571426\pi$
$632$		0			0
$633$	−0.585786	+	0.100505i	−0.0232829	+	0.00399472i
$634$		0			0
$635$	−6.82843			−0.270978
$636$		0			0
$637$	−3.41421			−0.135276
$638$		0			0
$639$	−0.970563	+	0.343146i	−0.0383949	+	0.0135746i
$640$		0			0
$641$			4.34315i			0.171544i	0.996315	+	0.0857720i	$0.0273357\pi$
$641$			4.34315i			0.171544i	−0.996315	+	0.0857720i	$0.972664\pi$
$642$		0			0
$643$			37.7574i			1.48900i	0.667620	+	0.744502i	$0.267313\pi$
$643$			37.7574i			1.48900i	−0.667620	+	0.744502i	$0.732687\pi$
$644$		0			0
$645$	10.4853	+	61.1127i	0.412858	+	2.40631i
$646$		0			0
$647$	−21.4558			−0.843516			−0.421758	−	0.906708i	$-0.638587\pi$
$647$	−21.4558			−0.843516			−0.421758	+	0.906708i	$0.638587\pi$
$648$		0			0
$649$	−12.4853			−0.490090
$650$		0			0
$651$	0.343146	+	2.00000i	0.0134489	+	0.0783862i
$652$		0			0
$653$		−	2.48528i		−	0.0972566i	−0.998817	−	0.0486283i	$-0.984515\pi$
$653$		−	2.48528i		−	0.0972566i	0.998817	−	0.0486283i	$-0.0154850\pi$
$654$		0			0
$655$		−	73.5980i		−	2.87571i
$656$		0			0
$657$	8.97056	−	3.17157i	0.349975	−	0.123735i
$658$		0			0
$659$	14.9706			0.583170			0.291585	−	0.956545i	$-0.405817\pi$
$659$	14.9706			0.583170			0.291585	+	0.956545i	$0.405817\pi$
$660$		0			0
$661$	40.1838			1.56297			0.781484	−	0.623926i	$-0.214463\pi$
$661$	40.1838			1.56297			0.781484	+	0.623926i	$0.214463\pi$
$662$		0			0
$663$	28.1421	−	4.82843i	1.09295	−	0.187521i
$664$		0			0
$665$			21.3137i			0.826510i
$666$		0			0
$667$			19.3137i			0.747830i
$668$		0			0
$669$	4.82843	−	0.828427i	0.186678	−	0.0320288i
$670$		0			0
$671$	−28.4853			−1.09966
$672$		0			0
$673$	2.00000			0.0770943			0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000			0.0770943			0.0385472	+	0.999257i	$0.487727\pi$
$674$		0			0
$675$	−16.8787	−	30.1924i	−0.649661	−	1.16211i
$676$		0			0
$677$		−	4.87006i		−	0.187172i	−0.995611	−	0.0935858i	$-0.970167\pi$
$677$		−	4.87006i		−	0.187172i	0.995611	−	0.0935858i	$-0.0298329\pi$
$678$		0			0
$679$		−	5.31371i		−	0.203921i
$680$		0			0
$681$	−7.00000	−	40.7990i	−0.268241	−	1.56342i
$682$		0			0
$683$	−29.5147			−1.12935			−0.564675	−	0.825314i	$-0.690998\pi$
$683$	−29.5147			−1.12935			−0.564675	+	0.825314i	$0.690998\pi$
$684$		0			0
$685$	−38.6274			−1.47588
$686$		0			0
$687$	0.313708	+	1.82843i	0.0119687	+	0.0697588i
$688$		0			0
$689$			45.4558i			1.73173i
$690$		0			0
$691$		−	22.0416i		−	0.838503i	−0.907870	−	0.419252i	$-0.862293\pi$
$691$		−	22.0416i		−	0.838503i	0.907870	−	0.419252i	$-0.137707\pi$
$692$		0			0
$693$	−2.00000	−	5.65685i	−0.0759737	−	0.214886i
$694$		0			0
$695$	6.00000			0.227593
$696$		0			0
$697$	48.2843			1.82890
$698$		0			0
$699$	15.3137	−	2.62742i	0.579218	−	0.0993780i
$700$		0			0
$701$			28.1421i			1.06291i	0.847085	+	0.531457i	$0.178355\pi$
$701$			28.1421i			1.06291i	−0.847085	+	0.531457i	$0.821645\pi$
$702$		0			0
$703$		−	5.17157i		−	0.195050i
$704$		0			0
$705$	−9.65685	+	1.65685i	−0.363698	+	0.0624007i
$706$		0			0
$707$	−18.2426			−0.686085
$708$		0			0
$709$	5.51472			0.207110			0.103555	−	0.994624i	$-0.466978\pi$
$709$	5.51472			0.207110			0.103555	+	0.994624i	$0.466978\pi$
$710$		0			0
$711$	4.00000	+	11.3137i	0.150012	+	0.424297i
$712$		0			0
$713$		−	4.68629i		−	0.175503i
$714$		0			0
$715$		−	23.3137i		−	0.871883i
$716$		0			0
$717$	−3.31371	−	19.3137i	−0.123753	−	0.721284i
$718$		0			0
$719$	20.6863			0.771468			0.385734	−	0.922610i	$-0.373948\pi$
$719$	20.6863			0.771468			0.385734	+	0.922610i	$0.373948\pi$
$720$		0			0
$721$	−16.9706			−0.632017
$722$		0			0
$723$	−1.07107	−	6.24264i	−0.0398334	−	0.232166i
$724$		0			0
$725$		−	32.1421i		−	1.19373i
$726$		0			0
$727$		−	10.6274i		−	0.394149i	−0.980389	−	0.197075i	$-0.936856\pi$
$727$		−	10.6274i		−	0.394149i	0.980389	−	0.197075i	$-0.0631441\pi$
$728$		0			0
$729$	−14.1421	+	23.0000i	−0.523783	+	0.851852i
$730$		0			0
$731$	−50.6274			−1.87252
$732$		0			0
$733$	0.384776			0.0142120			0.00710602	−	0.999975i	$-0.497738\pi$
$733$	0.384776			0.0142120			0.00710602	+	0.999975i	$0.497738\pi$
$734$		0			0
$735$	−5.82843	+	1.00000i	−0.214985	+	0.0368856i
$736$		0			0
$737$			18.6274i			0.686150i
$738$		0			0
$739$		−	11.8579i		−	0.436199i	−0.975927	−	0.218099i	$-0.930014\pi$
$739$		−	11.8579i		−	0.436199i	0.975927	−	0.218099i	$-0.0699857\pi$
$740$		0			0
$741$	36.3848	−	6.24264i	1.33663	−	0.229329i
$742$		0			0
$743$	−21.9411			−0.804942			−0.402471	−	0.915433i	$-0.631849\pi$
$743$	−21.9411			−0.804942			−0.402471	+	0.915433i	$0.631849\pi$
$744$		0			0
$745$	−26.1421			−0.957774
$746$		0			0
$747$	32.2843	−	11.4142i	1.18122	−	0.417624i
$748$		0			0
$749$			14.4853i			0.529281i
$750$		0			0
$751$			30.2843i			1.10509i	0.833483	+	0.552544i	$0.186343\pi$
$751$			30.2843i			1.10509i	−0.833483	+	0.552544i	$0.813657\pi$
$752$		0			0
$753$	−6.02944	−	35.1421i	−0.219725	−	1.28065i
$754$		0			0
$755$	−61.4558			−2.23661
$756$		0			0
$757$	−49.1127			−1.78503			−0.892516	−	0.451016i	$-0.851062\pi$
$757$	−49.1127			−1.78503			−0.892516	+	0.451016i	$0.851062\pi$
$758$		0			0
$759$	2.34315	+	13.6569i	0.0850508	+	0.495712i
$760$		0			0
$761$			1.02944i			0.0373171i	0.999826	+	0.0186585i	$0.00593954\pi$
$761$			1.02944i			0.0373171i	−0.999826	+	0.0186585i	$0.994060\pi$
$762$		0			0
$763$			2.48528i			0.0899732i
$764$		0			0
$765$	46.6274	−	16.4853i	1.68582	−	0.596027i
$766$		0			0
$767$	21.3137			0.769593
$768$		0			0
$769$	8.34315			0.300862			0.150431	−	0.988621i	$-0.451934\pi$
$769$	8.34315			0.300862			0.150431	+	0.988621i	$0.451934\pi$
$770$		0			0
$771$	−24.7279	+	4.24264i	−0.890554	+	0.152795i
$772$		0			0
$773$		−	1.27208i		−	0.0457535i	−0.999738	−	0.0228767i	$-0.992717\pi$
$773$		−	1.27208i		−	0.0457535i	0.999738	−	0.0228767i	$-0.00728253\pi$
$774$		0			0
$775$			7.79899i			0.280148i
$776$		0			0
$777$	1.41421	−	0.242641i	0.0507346	−	0.00870469i
$778$		0			0
$779$	62.4264			2.23666
$780$		0			0
$781$	−0.686292			−0.0245574
$782$		0			0
$783$	−21.8995	+	12.2426i	−0.782624	+	0.437516i
$784$		0			0
$785$			7.65685i			0.273285i
$786$		0			0
$787$			33.7574i			1.20332i	0.798752	+	0.601660i	$0.205494\pi$
$787$			33.7574i			1.20332i	−0.798752	+	0.601660i	$0.794506\pi$
$788$		0			0
$789$	−0.384776	−	2.24264i	−0.0136984	−	0.0798401i
$790$		0			0
$791$	−5.31371			−0.188934
$792$		0			0
$793$	48.6274			1.72681
$794$		0			0
$795$	13.3137	+	77.5980i	0.472189	+	2.75212i
$796$		0			0
$797$		−	12.1005i		−	0.428622i	−0.976765	−	0.214311i	$-0.931249\pi$
$797$		−	12.1005i		−	0.428622i	0.976765	−	0.214311i	$-0.0687506\pi$
$798$		0			0
$799$		−	8.00000i		−	0.283020i
$800$		0			0
$801$	−7.65685	−	21.6569i	−0.270542	−	0.765207i
$802$		0			0
$803$	6.34315			0.223845
$804$		0			0
$805$	13.6569			0.481341
$806$		0			0
$807$	31.9706	−	5.48528i	1.12542	−	0.193091i
$808$		0			0
$809$		−	10.6863i		−	0.375710i	−0.982197	−	0.187855i	$-0.939846\pi$
$809$		−	10.6863i		−	0.375710i	0.982197	−	0.187855i	$-0.0601535\pi$
$810$		0			0
$811$		−	29.5563i		−	1.03786i	−0.854816	−	0.518932i	$-0.826330\pi$
$811$		−	29.5563i		−	1.03786i	0.854816	−	0.518932i	$-0.173670\pi$
$812$		0			0
$813$	−21.3137	+	3.65685i	−0.747504	+	0.128251i
$814$		0			0
$815$	27.7990			0.973756
$816$		0			0
$817$	−65.4558			−2.29001
$818$		0			0
$819$	3.41421	+	9.65685i	0.119302	+	0.337438i
$820$		0			0
$821$		−	26.9706i		−	0.941279i	−0.882326	−	0.470640i	$-0.844023\pi$
$821$		−	26.9706i		−	0.941279i	0.882326	−	0.470640i	$-0.155977\pi$
$822$		0			0
$823$			32.9706i			1.14928i	0.818406	+	0.574641i	$0.194858\pi$
$823$			32.9706i			1.14928i	−0.818406	+	0.574641i	$0.805142\pi$
$824$		0			0
$825$	−3.89949	−	22.7279i	−0.135763	−	0.791285i
$826$		0			0
$827$	−3.45584			−0.120171			−0.0600857	−	0.998193i	$-0.519137\pi$
$827$	−3.45584			−0.120171			−0.0600857	+	0.998193i	$0.519137\pi$
$828$		0			0
$829$	−24.3848			−0.846918			−0.423459	−	0.905915i	$-0.639184\pi$
$829$	−24.3848			−0.846918			−0.423459	+	0.905915i	$0.639184\pi$
$830$		0			0
$831$	0.384776	+	2.24264i	0.0133477	+	0.0777963i
$832$		0			0
$833$		−	4.82843i		−	0.167295i
$834$		0			0
$835$			36.9706i			1.27942i
$836$		0			0
$837$	5.31371	−	2.97056i	0.183669	−	0.102678i
$838$		0			0
$839$	−1.17157			−0.0404472			−0.0202236	−	0.999795i	$-0.506438\pi$
$839$	−1.17157			−0.0404472			−0.0202236	+	0.999795i	$0.506438\pi$
$840$		0			0
$841$	5.68629			0.196079
$842$		0			0
$843$	−15.3137	+	2.62742i	−0.527432	+	0.0904930i
$844$		0			0
$845$		−	4.58579i		−	0.157756i
$846$		0			0
$847$			7.00000i			0.240523i
$848$		0			0
$849$	−17.4853	+	3.00000i	−0.600093	+	0.102960i
$850$		0			0
$851$	−3.31371			−0.113592
$852$		0			0
$853$	4.58579			0.157014			0.0785072	−	0.996914i	$-0.474985\pi$
$853$	4.58579			0.157014			0.0785072	+	0.996914i	$0.474985\pi$
$854$		0			0
$855$	60.2843	−	21.3137i	2.06168	−	0.728913i
$856$		0			0
$857$			38.0000i			1.29806i	0.760765	+	0.649028i	$0.224824\pi$
$857$			38.0000i			1.29806i	−0.760765	+	0.649028i	$0.775176\pi$
$858$		0			0
$859$		−	24.5858i		−	0.838856i	−0.907789	−	0.419428i	$-0.862231\pi$
$859$		−	24.5858i		−	0.838856i	0.907789	−	0.419428i	$-0.137769\pi$
$860$		0			0
$861$	2.92893	+	17.0711i	0.0998177	+	0.581780i
$862$		0			0
$863$	11.3726			0.387127			0.193564	−	0.981088i	$-0.437995\pi$
$863$	11.3726			0.387127			0.193564	+	0.981088i	$0.437995\pi$
$864$		0			0
$865$	32.6274			1.10937
$866$		0			0
$867$	1.84924	+	10.7782i	0.0628036	+	0.366046i
$868$		0			0
$869$			8.00000i			0.271381i
$870$		0			0
$871$		−	31.7990i		−	1.07747i
$872$		0			0
$873$	−15.0294	+	5.31371i	−0.508669	+	0.179842i
$874$		0			0
$875$	−5.65685			−0.191237
$876$		0			0
$877$	−57.7990			−1.95173			−0.975867	−	0.218368i	$-0.929927\pi$
$877$	−57.7990			−1.95173			−0.975867	+	0.218368i	$0.929927\pi$
$878$		0			0
$879$	2.17157	−	0.372583i	0.0732453	−	0.0125669i
$880$		0			0
$881$		−	7.85786i		−	0.264738i	−0.991200	−	0.132369i	$-0.957742\pi$
$881$		−	7.85786i		−	0.264738i	0.991200	−	0.132369i	$-0.0422584\pi$
$882$		0			0
$883$		−	9.31371i		−	0.313431i	−0.987644	−	0.156716i	$-0.949909\pi$
$883$		−	9.31371i		−	0.313431i	0.987644	−	0.156716i	$-0.0500906\pi$
$884$		0			0
$885$	36.3848	−	6.24264i	1.22306	−	0.209844i
$886$		0			0
$887$	2.82843			0.0949693			0.0474846	−	0.998872i	$-0.484879\pi$
$887$	2.82843			0.0949693			0.0474846	+	0.998872i	$0.484879\pi$
$888$		0			0
$889$	2.00000			0.0670778
$890$		0			0
$891$	−14.0000	+	11.3137i	−0.469018	+	0.379023i
$892$		0			0
$893$		−	10.3431i		−	0.346120i
$894$		0			0
$895$		−	22.1421i		−	0.740130i
$896$		0			0
$897$	−4.00000	−	23.3137i	−0.133556	−	0.778422i
$898$		0			0
$899$	5.65685			0.188667
$900$		0			0
$901$	−64.2843			−2.14162
$902$		0			0
$903$	−3.07107	−	17.8995i	−0.102199	−	0.595658i
$904$		0			0
$905$			69.5980i			2.31352i
$906$		0			0
$907$			37.5980i			1.24842i	0.781256	+	0.624210i	$0.214579\pi$
$907$			37.5980i			1.24842i	−0.781256	+	0.624210i	$0.785421\pi$
$908$		0			0
$909$	18.2426	+	51.5980i	0.605070	+	1.71140i
$910$		0			0
$911$	16.9706			0.562260			0.281130	−	0.959670i	$-0.409291\pi$
$911$	16.9706			0.562260			0.281130	+	0.959670i	$0.409291\pi$
$912$		0			0
$913$	22.8284			0.755511
$914$		0			0
$915$	83.0122	−	14.2426i	2.74430	−	0.470847i
$916$		0			0
$917$			21.5563i			0.711853i
$918$		0			0
$919$		−	29.6569i		−	0.978289i	−0.872203	−	0.489145i	$-0.837309\pi$
$919$		−	29.6569i		−	0.978289i	0.872203	−	0.489145i	$-0.162691\pi$
$920$		0			0
$921$	16.3137	−	2.79899i	0.537555	−	0.0922299i
$922$		0			0
$923$	1.17157			0.0385628
$924$		0			0
$925$	5.51472			0.181323
$926$		0			0
$927$	16.9706	+	48.0000i	0.557386	+	1.57653i
$928$		0			0
$929$		−	33.7990i		−	1.10891i	−0.832214	−	0.554454i	$-0.812927\pi$
$929$		−	33.7990i		−	1.10891i	0.832214	−	0.554454i	$-0.187073\pi$
$930$		0			0
$931$		−	6.24264i		−	0.204594i
$932$		0			0
$933$	−5.79899	−	33.7990i	−0.189850	−	1.10653i
$934$		0			0
$935$	32.9706			1.07825
$936$		0			0
$937$	21.5147			0.702855			0.351428	−	0.936215i	$-0.385696\pi$
$937$	21.5147			0.702855			0.351428	+	0.936215i	$0.385696\pi$
$938$		0			0
$939$	2.58579	+	15.0711i	0.0843840	+	0.491826i
$940$		0			0
$941$			15.6985i			0.511756i	0.966709	+	0.255878i	$0.0823645\pi$
$941$			15.6985i			0.511756i	−0.966709	+	0.255878i	$0.917635\pi$
$942$		0			0
$943$		−	40.0000i		−	1.30258i
$944$		0			0
$945$	8.65685	+	15.4853i	0.281607	+	0.503736i
$946$		0			0
$947$	57.3137			1.86245			0.931223	−	0.364451i	$-0.118743\pi$
$947$	57.3137			1.86245			0.931223	+	0.364451i	$0.118743\pi$
$948$		0			0
$949$	−10.8284			−0.351506
$950$		0			0
$951$	17.0711	−	2.92893i	0.553567	−	0.0949771i
$952$		0			0
$953$			0.686292i			0.0222312i	0.999938	+	0.0111156i	$0.00353827\pi$
$953$			0.686292i			0.0222312i	−0.999938	+	0.0111156i	$0.996462\pi$
$954$		0			0
$955$			12.4853i			0.404014i
$956$		0			0
$957$	−16.4853	+	2.82843i	−0.532893	+	0.0914301i
$958$		0			0
$959$	11.3137			0.365339
$960$		0			0
$961$	29.6274			0.955723
$962$		0			0
$963$	40.9706	−	14.4853i	1.32026	−	0.466782i
$964$		0			0
$965$			60.2843i			1.94062i
$966$		0			0
$967$			27.6569i			0.889384i	0.895683	+	0.444692i	$0.146687\pi$
$967$			27.6569i			0.889384i	−0.895683	+	0.444692i	$0.853313\pi$
$968$		0			0
$969$	8.82843	+	51.4558i	0.283610	+	1.65300i
$970$		0			0
$971$	−43.6985			−1.40235			−0.701176	−	0.712989i	$-0.747341\pi$
$971$	−43.6985			−1.40235			−0.701176	+	0.712989i	$0.747341\pi$
$972$		0			0
$973$	−1.75736			−0.0563384
$974$		0			0
$975$	6.65685	+	38.7990i	0.213190	+	1.24256i
$976$		0			0
$977$		−	44.0000i		−	1.40768i	−0.710356	−	0.703842i	$-0.751466\pi$
$977$		−	44.0000i		−	1.40768i	0.710356	−	0.703842i	$-0.248534\pi$
$978$		0			0
$979$		−	15.3137i		−	0.489428i
$980$		0			0
$981$	7.02944	−	2.48528i	0.224433	−	0.0793489i
$982$		0			0
$983$	−17.4558			−0.556755			−0.278377	−	0.960472i	$-0.589797\pi$
$983$	−17.4558			−0.556755			−0.278377	+	0.960472i	$0.589797\pi$
$984$		0			0
$985$	88.7696			2.82843
$986$		0			0
$987$	2.82843	−	0.485281i	0.0900298	−	0.0154467i
$988$		0			0
$989$			41.9411i			1.33365i
$990$		0			0
$991$			45.9411i			1.45937i	0.683785	+	0.729684i	$0.260333\pi$
$991$			45.9411i			1.45937i	−0.683785	+	0.729684i	$0.739667\pi$
$992$		0			0
$993$	−8.24264	+	1.41421i	−0.261572	+	0.0448787i
$994$		0			0
$995$	−24.9706			−0.791620
$996$		0			0
$997$	9.95837			0.315385			0.157692	−	0.987488i	$-0.449595\pi$
$997$	9.95837			0.315385			0.157692	+	0.987488i	$0.449595\pi$
$998$		0			0
$999$	−2.10051	−	3.75736i	−0.0664570	−	0.118878i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.h.a.575.3	✓	4
3.2	odd	2		672.2.h.d.575.1	yes	4
4.3	odd	2		672.2.h.d.575.2	yes	4
8.3	odd	2		1344.2.h.a.575.3		4
8.5	even	2		1344.2.h.e.575.2		4
12.11	even	2	inner	672.2.h.a.575.4	yes	4
24.5	odd	2		1344.2.h.a.575.4		4
24.11	even	2		1344.2.h.e.575.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.h.a.575.3	✓	4	1.1	even	1	trivial
672.2.h.a.575.4	yes	4	12.11	even	2	inner
672.2.h.d.575.1	yes	4	3.2	odd	2
672.2.h.d.575.2	yes	4	4.3	odd	2
1344.2.h.a.575.3		4	8.3	odd	2
1344.2.h.a.575.4		4	24.5	odd	2
1344.2.h.e.575.1		4	24.11	even	2
1344.2.h.e.575.2		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.292893 - 1.70711i) q^{3} +3.41421i q^{5} -1.00000i q^{7} +(-2.82843 + 1.00000i) q^{9} +O(q^{10})\)	\(q+(-0.292893 - 1.70711i) q^{3} +3.41421i q^{5} -1.00000i q^{7} +(-2.82843 + 1.00000i) q^{9} -2.00000 q^{11} +3.41421 q^{13} +(5.82843 - 1.00000i) q^{15} +4.82843i q^{17} +6.24264i q^{19} +(-1.70711 + 0.292893i) q^{21} +4.00000 q^{23} -6.65685 q^{25} +(2.53553 + 4.53553i) q^{27} +4.82843i q^{29} -1.17157i q^{31} +(0.585786 + 3.41421i) q^{33} +3.41421 q^{35} -0.828427 q^{37} +(-1.00000 - 5.82843i) q^{39} -10.0000i q^{41} +10.4853i q^{43} +(-3.41421 - 9.65685i) q^{45} -1.65685 q^{47} -1.00000 q^{49} +(8.24264 - 1.41421i) q^{51} +13.3137i q^{53} -6.82843i q^{55} +(10.6569 - 1.82843i) q^{57} +6.24264 q^{59} +14.2426 q^{61} +(1.00000 + 2.82843i) q^{63} +11.6569i q^{65} -9.31371i q^{67} +(-1.17157 - 6.82843i) q^{69} +0.343146 q^{71} -3.17157 q^{73} +(1.94975 + 11.3640i) q^{75} +2.00000i q^{77} -4.00000i q^{79} +(7.00000 - 5.65685i) q^{81} -11.4142 q^{83} -16.4853 q^{85} +(8.24264 - 1.41421i) q^{87} +7.65685i q^{89} -3.41421i q^{91} +(-2.00000 + 0.343146i) q^{93} -21.3137 q^{95} +5.31371 q^{97} +(5.65685 - 2.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3}+O(q^{10}) \) 4 * q - 4 * q^3	\( 4 q - 4 q^{3} - 8 q^{11} + 8 q^{13} + 12 q^{15} - 4 q^{21} + 16 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{35} + 8 q^{37} - 4 q^{39} - 8 q^{45} + 16 q^{47} - 4 q^{49} + 16 q^{51} + 20 q^{57} + 8 q^{59} + 40 q^{61} + 4 q^{63} - 16 q^{69} + 24 q^{71} - 24 q^{73} - 12 q^{75} + 28 q^{81} - 40 q^{83} - 32 q^{85} + 16 q^{87} - 8 q^{93} - 40 q^{95} - 24 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^11 + 8 * q^13 + 12 * q^15 - 4 * q^21 + 16 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^33 + 8 * q^35 + 8 * q^37 - 4 * q^39 - 8 * q^45 + 16 * q^47 - 4 * q^49 + 16 * q^51 + 20 * q^57 + 8 * q^59 + 40 * q^61 + 4 * q^63 - 16 * q^69 + 24 * q^71 - 24 * q^73 - 12 * q^75 + 28 * q^81 - 40 * q^83 - 32 * q^85 + 16 * q^87 - 8 * q^93 - 40 * q^95 - 24 * q^97

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